Astronomy Principles and Practice Fourth Edition.pdf

Spectral lines 223
as the orbits are quantized, so must be the amounts of radiation that are involved. In other words, the
radiation emitted or absorbed by atoms is quantized.
Following Bohr’s postulate, equation (15.18) may be written as
( )
nh
2π = e me r 1/2
4πε 0
where n has any integral value (n = 1, 2, 3 ...).
The radii of the Bohr orbits can, therefore, be expressed as
r = ε 0n 2 h 2
πe 2 (15.19)
m e
and by putting n = 1 and inserting the values of ε 0 , h, e and m e , the size of the smallest Bohr orbit is
obtained, giving a value equal to 5·3 × 10 −11 mor0·53 Å.
The energies of the Bohr orbits are given by substituting equation (15.19) into (15.17) which gives
E n =−
8ε0 2n2 h 2 . (15.20)
The energy of the first orbit, known as the ground state, is obtained by letting n = 1, giving a value
equal to −2·17 × 10 −18 J.
It is more convenient to describe the energies of electron orbits in units of electron volts (eV)
rather than in joules. One electron volt is defined as the energy acquired by an electron after it has
been accelerated by a potential difference of 1 volt. Thus,
e4 m e
1eV= charge × potential difference.
Now the charge on an electron is 1·6 × 10 −19 C and, therefore,
1eV= 1·6 × 10 −19 J.
The energy of the ground state in the hydrogen atom in units of eV is given by
2·17 × 10−18
−
1·5 × 10 −19 eV =−13·6eV
and the energies of the other orbits or excited states are, therefore, given by
E n = −13·6
n 2 eV. (15.21)
Energy levels are frequently depicted in a pictorial way as a series of lines—the energy levels for the
hydrogen atom are illustrated in figure 15.7.
15.7.3 The hydrogen spectrum
By setting up this model of the hydrogen atom involving the quantization of the orbital energies, Bohr
succeeded in providing an explanation of emission line spectra, in particular the hydrogen spectrum
which had been found by Balmer in 1885 to be positioned according to a fairly simple formula. Bohr
suggested that a change of energy level, or transition, corresponds to the absorption or emission of a
quantum of radiation whose energy, hν, is given by Planck’s formula. Thus, if two energy levels are
designated by E m and E n , their energy difference is given by
E = E m − E n = hν.